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Properties

Label 2-1380-69.68-c1-0-22
Degree $2$
Conductor $1380$
Sign $0.800 + 0.598i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0831 + 1.73i)3-s + 5-s − 4.02i·7-s + (−2.98 + 0.287i)9-s − 0.641·11-s − 2.89·13-s + (0.0831 + 1.73i)15-s + 3.10·17-s − 1.33i·19-s + (6.95 − 0.334i)21-s + (3.05 − 3.69i)23-s + 25-s + (−0.746 − 5.14i)27-s − 3.14i·29-s + 6.11·31-s + ⋯
L(s)  = 1  + (0.0480 + 0.998i)3-s + 0.447·5-s − 1.51i·7-s + (−0.995 + 0.0959i)9-s − 0.193·11-s − 0.804·13-s + (0.0214 + 0.446i)15-s + 0.752·17-s − 0.306i·19-s + (1.51 − 0.0729i)21-s + (0.636 − 0.771i)23-s + 0.200·25-s + (−0.143 − 0.989i)27-s − 0.583i·29-s + 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.800 + 0.598i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.800 + 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.529643524\)
\(L(\frac12)\) \(\approx\) \(1.529643524\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.0831 - 1.73i)T \)
5 \( 1 - T \)
23 \( 1 + (-3.05 + 3.69i)T \)
good7 \( 1 + 4.02iT - 7T^{2} \)
11 \( 1 + 0.641T + 11T^{2} \)
13 \( 1 + 2.89T + 13T^{2} \)
17 \( 1 - 3.10T + 17T^{2} \)
19 \( 1 + 1.33iT - 19T^{2} \)
29 \( 1 + 3.14iT - 29T^{2} \)
31 \( 1 - 6.11T + 31T^{2} \)
37 \( 1 + 7.06iT - 37T^{2} \)
41 \( 1 + 3.72iT - 41T^{2} \)
43 \( 1 + 1.20iT - 43T^{2} \)
47 \( 1 - 3.57iT - 47T^{2} \)
53 \( 1 - 5.12T + 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 - 0.921iT - 61T^{2} \)
67 \( 1 - 1.27iT - 67T^{2} \)
71 \( 1 - 1.44iT - 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 3.08iT - 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + 4.31T + 89T^{2} \)
97 \( 1 + 13.1iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.727341627199429020505595533521, −8.861223149632572681402769068364, −7.86503108895860049532382643976, −7.13372791438042977747296195396, −6.12862973708058565898273996396, −5.07808447615038734395815432465, −4.43325718649041947911500106689, −3.52578975940688864041215175835, −2.46726537295204734987597755043, −0.64724620232520265309531002584, 1.38076995881213313114552729180, 2.47537043937397495535208656886, 3.11724782392929530097071625159, 4.97218852268920953650962087404, 5.60321957206022090834311069873, 6.31640265934313971935473165591, 7.23525659067549725919087178353, 8.081193500909035813763254277182, 8.777545952485105241125170991168, 9.515198925905106257887951965166

Graph of the $Z$-function along the critical line